The covariant Lyapunov tensor and the Lyapunov stability with respect to time-dependent Riemannian metrics

TL;DR

This paper demonstrates that Lyapunov stability of solutions to first-order dynamic equations can be achieved through the selection of suitable time-dependent Riemannian metrics, offering a flexible approach to stability analysis.

Contribution

It introduces a method to ensure Lyapunov stability by choosing appropriate time-dependent Riemannian metrics for dynamic equations.

Findings

Lyapunov stability can be controlled via Riemannian metrics.

Any solution can be made Lyapunov stable with the right metric.

The approach provides a new perspective on stability analysis.

Abstract

We show that any solution of a smooth first order dynamic equation can be made Lyapunov stable at will by the choice of an appropriate time-dependent Riemannian metric.